Let R be a relation defined on the set 1,2,3,4 1,2,3,4 by R = ((a, b),(c, d)): 2 a+3 b=3 c+4 d . Then the number of elements in R is

Step-by-step solution

We need 2a+3b = 3c+4d where a,b,c,d 1,2,3,4 . Possible values of 2a+3b : range from 5 to 20 . Possible values of 3c+4d : range from 7 to 28 . For each common value k , the number of elements in R equals (number of (a,b) with 2a+3b=k ) (number of (c,d) with 3c+4d=k ). k=7 : (a,b)=(2,1) , (c,d)=(1,1) 1 1=1 k=10 : (2,2) , (2,1) 1 k=11 : (1,3),(4,1) , (1,2) 2 k=13 : (2,3) , (3,1) 1 k=14 : (1,4),(4,2) , (2,2) 2 k=15 : (3,3) , (1,3) 1 k=16 : (2,4) , (4,1) 1 k=17 : (4,3) , (3,2) 1 k=18 : (3,4) , (2,3) 1 k=20 : (4,4) , (4,